A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of Quasi-Nonexpansive Mappings

The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S.Takahashi and W. Takahashi (2008), and many others.

Throughout this paper, we assume that is a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively.

Recall that the following definitions.

(1)A mapping is said to be nonexpansive if

(2)A mapping is said to be quasi-nonexpansive if

We denote be the set of fixed points of .

Let be a single-valued nonlinear mapping and to a set-valued mapping. The variational inclusion problem is to find such that

where is the zero vector in . The set of solutions of problem (1.3) is denoted by .

Definition 1.1.

A mapping is said to be a -inverse-strongly monotone if there exists a constant with the property

Remark 1.2.

It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous. It is easy to see that if any constant is in , then the mapping is nonexpansive, where is the identity mapping on

A set-valued mapping is called monotone if for all , and implying . A monotone mapping; is maximal if its graph of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for for all imply .

Definition 1.3.

Let be a set-valued maximal monotone mapping, then the single-valued mapping defined by

is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Remark 1.4.

(R1) The resolvent operator is single-valued and nonexpansive for all , that is,

(R2)The resolvent operator is 1-inverse strongly monotone; see [1], that is,

(R3)The solution of problem (1.3) is a fixed point of the operator for all ; see also [2], that is,

(R4)If , then the mapping is nonexpansive.

(R5) is closed and convex.

Let be a nonlinear mapping, let be a real-valued function and a bifunction from to . We consider the following generalized mixed equilibrium problem.

Finding such that

The set of such is denoted by that is,

It is easy to see that is solution of problem (1.9) implies that .

(i)In the case of (:the zero mapping), then the generalized mixed equilibrium problem (1.9) is reduced to the mixed equilibrium problem. Finding such that

The set of solution of (1.11) isdenoted by

(ii)In the case of , then the generalized mixed equilibrium problem (1.9) is reduced to the generalized equilibrium problem. Finding such that

The set of solution of (1.12) is denoted by

(iii)In the case of (:the zero mapping) and , then the generalized mixed equilibrium problem (1.9) is reduced to the equilibrium problem. Finding such that

The set of solution of (1.13) is denoted by

(iv)In the case of , and then the generalized mixed equilibrium problem (1.9) is reduced to the variational inequality problem. Finding such that

The set of solution of (1.14) is denoted by

The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problems as special cases (see, e.g., [3–8]). Some methods have been proposed to solve the generalized mixed equilibrium problems, generalized equilibrium problems and equilibrium problems; see, for instance, [9–22].

In 2007, Takahashi et al. [23] proved the following strong convergence theorem for a nonexpansive mapping by using the shrinking projection method in mathematical programming. For a and , they defined a sequence as follows

where . They proved that the sequence generated by (1.15) converges weakly to , where

In 2008, S. Takahashi and W. Takahashi [24] introduced the following iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Starting with arbitrary , define sequences , and by

They proved that under certain appropriate conditions imposed on , and , the sequence generated by (1.16) converges strongly to

In 2008, Zhang et al. [25] introduced the following new iterative scheme for finding a common element of the set of solutions to the problem (1.3) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitrary , define sequences and by

where is the resolvent operator associated with and a positive number is a sequence in the interval .

In 2008, Peng et al. [26] introduced the following iterative scheme by the viscosity approximation method for finding a common element of the set of solutions to the problem (1.3), the set of solutions of an equilibrium problems and the set of fixed points of nonexpansive mappings in a Hilbert space. Starting with an arbitrary , define sequences and by

They proved that under certain appropriate conditions imposed on and , the sequence generated by (1.18) converges strongly to

In 2010, Katchang and Kumam [27] introduced an iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings, and inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space.

In this paper, motivated and inspired by the previously mentioned results, we introduce an iterative scheme by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed shrinking projection method under some suitable conditions. The results obtained in this paper extend and improve several recent results in this area.

Let be a real Hilbert space and let be a nonempty closed convex subset of Recall that the (nearest point) projection from onto assigns to each the unique point in satisfying the property

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1.

For a given and ,

It is well known that is a firmly nonexpansive mapping of onto and satisfies

Lemma 2.2 (see [1]).

Let be a maximal monotone mapping and let be a Lipshitz continuous mapping. Then the mapping is a maximal monotone mapping.

Lemma 2.3 (see [28]).

Let be a closed convex subset of and let be a bounded sequence in . Assume that

(1)the weak -limit set ,

(2)for each , exists.

Then is weakly convergent to a point in .

Lemma 2.4 (see [29]).

Each Hilbert space satisfies Opial's condition, that is, for any sequence with , the inequality

holds for each with .

Lemma 2.5 (see [30]).

Each Hilbert space satisfies the Kadec-Klee property, that is, for any sequence with and together imply .

For solving the generalized equilibrium problems, let us give the following assumptions for and the set :

(A1) for all

(A2) is monotone, that is, for all

(A3)for each is weakly upper semicontinuous;

(A4)for each is convex and lower semicontinuous;

(B1)for each and there exists a bounded subset and such that for any

(B2)C is bounded set.

Lemma 2.6 (see [31]).

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let be a proper lower semicontinuous and convex function such that . For and , define a mapping as follows:

Assume that either or holds. Then, the following conclusions hold:

(1)for each

(2) is single-valued;

(3) is firmly nonexpansive, that is, for any

(4)

(5) is closed and convex.

Remark 2.7.

Replacing with in (2.5), then there exists , such that

In this section, we will introduce an iterative scheme by using shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of variational inclusion problems in a real Hilbert space.

Let be a finite family of nonexpansive mappings of into itself, and let be real numbers such that for every . We define a mapping as follows:

Such a mapping is called the K-mapping generated by and ; see [32].

We have the following crucial Lemma 3.1 and Lemma 3.2 concerning -mapping which can be found in [14]. Now we only need the following similar version in Hilbert spaces.

Lemma 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasi-nonexpansive mappings and -Lipschitz mappings of into itself with and let be real numbers such that for every , and . Let be the K-mapping generated by and . Then, the followings hold:

(1) is quasi-nonexpansive and Lipschitz,

(2).

Lemma 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a finite family of quasi-nonexpansive mappings and -Lipschitz mappings of into itself and sequences in such that Moreover, for every , let and be the K-mappings generated by and , and and , respectively. Then, for every , we have

Now we study the strong convergence theorem concerning the shrinking projection method.

Theorem 3.3.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself, and let be a -inverse-strongly monotone mapping of into , let a -inverse-strongly monotone mapping of into and be a maximal monotone mapping. Assume that

Let be the -mapping generated by and . Let , , , and be sequences generated by , , and let

where satisfy the following conditions:

(i) for some with ;

(ii), for some with ;

(iii) for some with .

Then, and converge strongly to

Proof.

In the light of the definition of the resolvent, can be rewritten as . Let and using the fact be a sequence of mappings defined as in Lemma 2.6, is an -inverse-strongly monotone and that , where for some with , we can write

Next, we will divide the proof into six steps.

Step 1.

We first show that is well defined and is closed and convex for any .

From the assumption, we see that is closed and convex. Suppose that is closed and convex for some . Next, we show that is closed and convex for some . For any , we obtain that

is equivalent to

Thus is closed and convex. Then, is closed and convex for any . This implies that is well defined.

Step 2.

Next, we show by induction that for each .

Taking and by condition (ii), we get that is nonexpansive for all . From the assumption, we see that . Suppose for some . For any , we have

Thus, we have

It follows that This implies that for each .

Step 3.

Next, we show that and .

From , we have

for each . Using we also have

So, for , we have

This implies that

From and , we obtain

From (3.13), we have, for

It follows that

Thus the sequence is a bounded and nonincreasing sequence, so exists, that is,

Indeed, from (3.13), we get

From (3.16), we obtain

Since we have

By (3.18), we obtain

Step 4.

Next, we show that

For any given , . It is easy to see that . As is nonexpansive, we have

Similarly, we can prove that

Observe that

Substituting (3.21) into (3.23), and using conditions (i) and (ii), we have

It follows that

Since , we obtain

Since the resolvent operator is 1-inverse strongly monotone, we obtain

which yields that

Similarly, we obtain

Substituting (3.28) into (3.23), and using condition (i), we have

It follows that

Applying and as to the last inequality, we get

Note that

Substituting (3.22) into (3.33), and using conditions (i) and (ii), we have

It follows that

Since , we obtain

Substituting (3.29) into (3.33), and using conditions (i) and (ii), we have

It follows that

Applying and as to the last inequality, we get

From (3.32) and (3.39), we have

From (3.33), (3.4), and condition (iii), we have

It follows that

Since , we obtain

On the other hand, in the light of Lemma 2.6(3), is firmly nonexpansvie, so we have

which implies that

Using (3.41) again and (3.45), we have

It follows from the condition (i) that

Since and , it is implied that

From (3.39) and (3.48), we have

By (3.3), we get

Since for some with , and as , we also have

From (3.40) and (3.48), we have

Furthermore, by the triangular inequality, we also have

Applying (3.51) and (3.52), we obtain

Let be the mapping defined by (3.1). Since is bounded, applying Lemma 3.2 and (3.54), we have

Step 5.

Next, we show that

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . Since and is closed and convex, is weakly closed and hence . From we obtain

(a) First, we prove that .

We observe that is a -Lipschitz monotone mapping and . From Lemma 2.2, we know that is maximal monotone. Let , that is, . Since , we have

that is,

By virtue of the maximal monotonicity of , we have

and so

It follows from , and that

It follows from the maximal monotonicity of that , that is, .

(b) Next, we show that . Since dom , we have

From (A2), we also have

And hence

For with and let Since and we have So, from (3.63), we have

Since we have . Further, from the inverse strongly monotonicity of we have So, from (A5), the weakly lower semicontinuity of and , we have

as From (A1), (A4) and (3.65), we also get

Letting we have, for each

This implies that

(c) Now, we prove that .

Assume Since and we know that and it follows by the Opial's condition (Lemma 2.4) that

which is a contradiction. Thus, we get .

The conclusion is .

Step 6.

Finally, we show that and , where

Since is nonempty closed convex subset of , there exists a unique such that Since and , we have

for all . From (3.69), is bounded, so . By the weak lower semicontinuity of the norm, we have

However, Since , we have

Using (3.69) and (3.70), we obtain . Thus and So, we have

Thus, we obtain that

From , we obtain . Using the Kadec-Klee property (Lemma 2.5) of , we obtain that

and hence in norm. Finally, noticing we also conclude that in norm. This completes the proof.

Corollary 3.4.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of nonexpansive mappings of into itself, let be a -inverse-strongly monotone mapping of into , let be a -inverse-strongly monotone mapping of into and a maximal monotone mapping. Assume that

Let be the -mapping generated by and . Let , , , and be sequences generated by (3.3) satisfying the following conditions in Theorem 3.3. Then, and converge strongly to

From Theorem 3.3, we can obtain the following results.

Theorem 3.5.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4), and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself, let be a -inverse-strongly monotone mapping of into and let be a -inverse-strongly monotone mapping of into . Assume that

Let be the -mapping generated by and . Let , , , and be sequences generated by , , and let

where satisfy the following conditions:

(i) for some with ;

(ii), for some with ;

(iii) for some with .

Then, and converge strongly to

Proof.

In Theorem 3.3 take , where is the indicator function of , that is,

Then the variational inclusion problem (1.3) is equivalent to variational inequality problem (1.14), that is, to find such that

Again, since , then

and so we have

We can obtain the desired conclusion from Theorem 3.3 immediately.

Next, we consider another class of important nonlinear mappings: strict pseudocontractions.

Definition 3.6.

A mapping is called strictly pseudocontractionif there exists a constant such that

If , then is nonexpansive.

In this case, let a -strictly pseudocontraction. Putting , then is a -inverse-strongly monotone mapping. In fact, from (3.82) we have

Observe that

Hence, we obtain

This shows that is -inverse-strongly monotone mapping.

Now, we get the following result.

Theorem 3.7.

Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function with assumption (B1) or (B2). Let be a finite family of quasi-nonexpansive and -Lipschitz mappings of into itself, let be a -strictly pseudocontraction mapping of into and let be a -strictly pseudocontraction mapping of into . Assume that

Let be the -mapping generated by and . Let , , , and be sequences generated by , , and let

where satisfy the following conditions:

(i) for some with ;

(ii), for some with ;

(iii) for some with .

Then, and converge strongly to

Proof.

Taking and , respectively. Then we see that is -inverse-strongly monotone and is -inverse-strongly monotone, respectively. We have and

By using Theorem 3.5, it is easy to obtain the desired conclusion.

The authors would like to express their thank to the referees for helpful suggestions. The first author was supported by the National Research Council of Thailand and the Faculty of Science and Technology RMUTT Research Fund. The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute. The third author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5380044.

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